8 Surveys on some other aspects of the calculus of claspers
8.6 Claspers and gropes
Some authors use gropes to study links and 3–manifolds [6] [29]. We explain here some relationships between claspers and gropesembeddedin 3–manifolds.
For the definitions of gropes and capped gropes, see [9]. We define a (capped) k–grope X for a link γ in M to be a (capped) grope X of class k embedded in M intersecting γ only by some transverse double points in the caps of X. (In the non-capped case, X and γ are disjoint.)
Two links γ and γ0 in M are said to be related by a(capped) k–gropingif there is a (capped)k–gropeX forγ and a band B connecting a component of γ and the bottom b of X in such a way that B∩X =∂B∩b and B∩γ =∂B∩γ, and if the band sum of γ and b along the band B is equivalent to γ0.
We can prove that two links in M are related by a sequence of capped k–
gropings (resp. k–gropings) if and only if they are Ck–equivalent (resp. Ak– equivalent). As corollaries to this, we can prove that anAk–move on a link inM preserves the homotopy classes of the components of a link up to thekth lower central series subgroup of π1M, and that the kth nilpotent quotient (ie, the quotient by the k+ 1st lower central series subgroup) of the fundamental group of the link exterior is an invariant ofAk–equivalence classes of links (and hence of Ck–equivalence classes). From this we can also prove that an Ak–surgery on a 3–manifold preserves the kth nilpotent quotient of the fundamental group of 3–manifolds.
Recall that for a knot γ in a 3–manifold M, the homotopy class of γ lies in the kth lower central series subgroup of π1M if and only if there is map f of a grope X of class k into M such that the bottom of X is mapped diffeomorphically onto γ. This condition is much weaker than that γ bounds an embedded k–grope in M. In some sense, embedded gropes, and hence tree and graph claspers, may be thought of as a kind of “geometric commutator” in a 3–manifold. Gropes thus provide us another way of thinking of calculus of claspers as a commutator calculus of a new kind.
References
[1] D Bar-Natan,On the Vassiliev knot invariants, Topology, 34 (1995) 423–472 [2] D Bar-Natan,Vassiliev homotopy string link invariants, J. Knot Theory
Ram-ifications, 4 (1995) 13–32
[3] J S Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28 (1993) 253–287
[4] J S Birman, X S Lin, Knot polynomials and Vassiliev’s invariants, Invent.
Math. 111 (1993) 225–270
[5] T D Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products, Mem. Amer. Math. Soc. 84 (1990) no. 427
[6] T D Cochran, A Gerges, K Orr, Surgery equivalence relations on three–
manifolds, preprint
[7] T D Cochran,P M Melvin,Finite type invariants of 3–manifolds, preprint [8] L Crane,D Yetter, On algebraic structures implicit in Topological Quantum
Field Theories, preprint
[9] M H Freedman,P Teichner, 4–Manifold topology II: Dwyer’s filtration and surgery kernels, Invent. Math. 122 (1995) 531–557
[10] S Garoufalidis, N Habegger, The Alexander polynomial and finite type 3–
manifold invariants, preprint
[11] S Garoufalidis, J Levine, Finite type 3–manifold invariants, the mapping class group and blinks, J. Diff. Geom. 47 (1997) 257–320
[12] S Garoufalidis,T Ohtsuki,On finite type 3–manifold invariants III: manifold weight systems, Topology, 37 (1998) 227–243
[13] M N Goussarov, A new form of the Conway–Jones polynomial of oriented links, from: “Topology of manifolds and varieties”, Adv. Soviet Math. 18, Amer.
Math. Soc. Providence, RI (1994) 167–172
[14] M N Goussarov, On n–equivalence of knots and invariants of finite degree, from: “Topology of manifolds and varieties”, Adv. Soviet Math. 18, Amer.
Math. Soc. Providence, RI (1994) 173–192
[15] M N Goussarov,Interdependent modifications of links and invariants of finite degree, Topology, 37 (1998) 595–602
[16] M N Goussarov,New theory of invariants of finite degree for 3–manifolds, (in Russian) preprint
[17] N Habegger, A computation of the universal quantum 3–manifold invariant for manifolds of rank greater than 2, preprint
[18] N Habegger,X S Lin,The classification of links up to link-homotopy, J. Amer.
Math. Soc. 3 (1990) 389–419
[19] N Habegger,G Masbaum,The Kontsevich integral and Milnor’s invariants, preprint
[20] K Habiro,Claspers and the Vassiliev skein modules, PhD thesis, University of Tokyo (1997)
[21] K Habiro,Clasp-pass moves on knots, unpublished
[22] R Hain,Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc.
10 (1997) 597–651
[23] D Johnson,An abelian quotient of the mapping class group, Math. Ann. 249 (1980) 225–242
[24] T Kerler, Genealogy of nonperturbative quantum-invariants of 3–manifolds:
The surgical family, from: “Geometry and physics”, Lecture Notes in Pure and Appl. Math. 184, Dekker, New York (1997) 503–547
[25] T Kerler, Bridged links and tangle presentations of cobordism categories, preprint
[26] R C Kirby,A calculus of framed links in S3, Invent. Math. 65 (1978) 35–56 [27] T Kohno, Vassiliev invariants and de-Rham complex on the space of knots,
from: “Symplectic geometry and quantization”, Contemp. Math. 179, Amer.
Math. Soc. Providence, RI (1994) 123–138
[28] M Kontsevich,Vassiliev’s knot invariants, from: “I M Gelfand seminar”, Adv.
Soviet Math. 16 Part 2, Amer. Math. Soc. Providence, RI (1993) 137–150 [29] V S Krushkal,Additivity properties of Milnor’s µ–invariants, J. Knot Theory¯
Ramifications, 7 (1998) 625–637
[30] T Q T Le, An invariant of integral homology 3–spheres which is universal for all finite type invariants, from: “Solitons, geometry and topology: on the cross-road”, Amer. Math. Soc. Transl. Ser. 2, 179 (1997) 75–100,
[31] T Q T Le, J Murakami, T Ohtsuki, On a universal quantum invariant of 3–manifolds, Topology, 37 (1998) 539–574
[32] X S Lin, Power series expansions and invariants of links, from “Geometric topology”, AMS/IP Stud. Adv. Math. 2.1, Amer. Math. Soc. Providence, RI (1997) 184–202
[33] S Majid,Algebras and Hopf algebras in braided categories, from: “Advances in Hopf algebras”, Lecture Notes in Pure and Appl. Math. 158, Dekker, New York (1994) 55–105
[34] S Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge (1995)
[35] S V Matveev,Generalized surgeries of three–dimensional manifolds and repre-sentations of homology spheres, (in Russian) Mat. Zametki 42 (1987) 268–278, 345
[36] J Milnor,Link groups, Ann. of Math. 59 (1954) 177–195
[37] J Milnor,Isotopy of links Algebraic geometry and topology, from: “A sympo-sium in honor of S Lefschetz”, Princeton University Press, Princeton, NJ (1957) 280–306
[38] S Morita,Casson’s invariant for homology 3–spheres and characteristic classes of surface bundles I, Topology, 28 (1989) 305–323
[39] S Morita,Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699–726
[40] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75–89
[41] K Y Ng,Groups of ribbon knots, Topology, 37 (1998) 441–458
[42] T Ohtsuki,Finite type invariants of integral homology 3–spheres, J. Knot The-ory Ramifications, 5 (1996) 101–115
[43] T Stanford, Finite type invariants of knots, links, and graphs, Topology, 35 (1996) 1027–1050
[44] T Stanford,Braid commutators and Vassiliev invariants, Pacific Jour. of Math.
174 (1996) 269–276
[45] T Stanford, Vassiliev invariants and knots modulo pure braid subgroups, preprint
[46] V A Vassiliev,Cohomology of knot spaces, from: “Theory of Singularities and its Applications”, Adv. Soviet Math., Amer. Math. Soc. Providence, RI (1990) 23–69